This article proposes a direct power control strategy based on bang-bang funnel control (BBFC-DPC) for the three-phase grid-connected voltage-source inverters. The BBFC has inherently robust nature due to its model-free design. It can provide time-optimal dynamics and effectively constrain the tracking error of output variables within the prespecified funnels. The BBFC-DPC approach is implemented based on a newly established DPC model in the abc frame, which enables satisfying decoupled power control without using coordinate transformations and feedforward decoupling terms. Built upon the merits of BBFC, the BBFC-DPC features high-quality static output performance, promoted dynamic responses, stronger robustness against parameter uncertainties, and pleasing resilience to slight grid disturbances. Moreover, an extended BBFC-DPC scheme equipped with reactive power support and current limitation functions is developed as a feasible alternative to tackle unbalanced network conditions in weak grid. Finally, both the simulation and experimental results are well in line with the theoretical expectations, which validates the effectiveness of the proposed method.